How to make an object move on a Mobius strip?
Create a Bezier Curve. Open it (it shouldn't be closed). Tilt one end 180°. &#9096; CtrlT
The Curve should resemble the twist of the moebius strip.

Add a Follow Path Constraint to the object. Check Fixed Position and set a key with the offset at 0 and 1.

Select the created keyframes and make them linear. T > L
Then make the graph cyclic. &#8679; ShiftE > Make Cyclic  
Set a key on the first frame of the rotation. I
Set a key after the second keyframe of the offset on the rotation with the object rotated 180°, making it ready for the second circulation.
Set the new keys to constant interpolation. T > C



Wheeee!